Can a Double Sequence Have Different Iterated Limits?

A double sequence is a collection of numbers $a_{m,n}$ indexed by two integers. You can ask what happens when both indices grow, or take limits one index at a time. Those are different questions: the two iterated limits may exist and disagree.

The standard example below has one iterated limit equal to $0$ and the other equal to $1$. That difference also shows the double sequence has no joint limit.

A worked example

For positive integers $m,n$, define

$$a_{m,n}=\frac{m}{m+n}.$$

First hold $m$ fixed and let $n$ tend to infinity. The denominator grows without bound while the numerator stays fixed, so

$$\lim_{n\to\infty}a_{m,n}=0.$$

Now take the limit of these row limits as $m$ tends to infinity. Since every row limit is $0$,

$$\lim_{m\to\infty}\left(\lim_{n\to\infty}a_{m,n}\right)=0.$$

Reverse the order. Hold $n$ fixed and let $m$ tend to infinity. Dividing numerator and denominator by $m$ gives

$$\lim_{m\to\infty}\frac{m}{m+n} =\lim_{m\to\infty}\frac{1}{1+n/m}=1.$$

The column limits are all $1$, so

$$\lim_{n\to\infty}\left(\lim_{m\to\infty}a_{m,n}\right)=1.$$

Thus the iterated limits exist but are unequal. The order of the limits matters.

What a joint limit means

A joint limit $a_{m,n}\to L$ as $m,n\to\infty$ means that for every $\varepsilon>0$, there are cutoffs $M,N$ such that $|a_{m,n}-L|<\varepsilon$ whenever $m\ge M$ and $n\ge N$. Both indices must be large, regardless of how quickly one grows compared with the other.

For the example, different paths give different values:

• Along $m=n$, $a_{n,n}=1/2$.
• Along $m=n^2$, $a_{n^2,n}=n/(n+1)\to1$.
• Along $n=m^2$, $a_{m,m^2}=1/(1+m)\to0$.

If a joint limit existed, every such path would have to approach that same limit. Since these paths give different results, there is no joint limit.

How the notions are related

A joint limit, when it exists, is stronger than merely having iterated limits. If the relevant inner limits exist for all indices and the joint limit is $L$, then each iterated limit is also $L$. But a joint limit does not, by itself, guarantee that every row or column has a limit: the joint condition only controls indices once both are sufficiently large, so a fixed row or column can behave differently.

Conversely, two unequal iterated limits rule out a joint limit. If the joint limit existed and the inner limits were defined, taking one index to infinity while the other was already beyond its cutoff would force the iterated limits to agree with the joint limit.

Common mistakes

• Confusing iterated limits with a joint limit. The expressions $\lim_m\lim_n a_{m,n}$ and $\lim_n\lim_m a_{m,n}$ describe ordered operations. They do not automatically describe the behavior as both indices grow together.
• Checking only the diagonal. Along $m=n$, the example gives $1/2$, but that single path does not establish a joint limit. Other paths give different values.
• Assuming the limits must commute. Interchanging limit operations requires suitable hypotheses; it is not valid for every double sequence.
• Ignoring existence. One or both iterated limits may fail to exist. In that case, there is no finite value to compare for that order.

So if the question is whether two sequences of limits associated with a double sequence can converge to different numbers, the answer is yes. The example $a_{m,n}=m/(m+n)$ gives iterated limits $0$ and $1$ in opposite orders.